Long-range Coulomb interactions see 3.8 O, Oe (r) 4IEokBTr Hydrogen bond interactions
Long-range Coulomb interactions see 3.8 Hydrogen bond interactions 5 2 1 2 0 4 B QQe U r SH k Tr
The Lennard-Jones or Hard Core potential The L-j(6-12) Potential is often used as an approximate model for a total (repulsion plus attraction van der Waals force as a function of distance E Repulsive +A/r12 U Attractive-B/r6 A B U=0 r>ro 12 6 U=∞T≤10
The Lennard-Jones or Hard Core Potential The L-J (6-12) Potential is often used as an approximate model for a total (repulsion plus attraction) van der Waals force as a function of distance. ܷ = 6 ܣ ݎ ଵଶ െ ܤ ݎ ݎ < ݎ 0ܷ = ݎ ݎ λܷ =
MayerfF-Function and Excluded Volume The Probability P(r of finding two monomers at distance r. Boltzmann factor P(r)a exp[-U (r)/(kBT) Mayer f-function difference of boltzmann factor for two monomers at r and at oo f(r)=exp (u (r/kB t)-1 0.5 Exclude volume v-I f(r)d3r=(1-exp[-U(r)/(kBT)Ddr
Mayer f-Function and Excluded Volume 7 The Probability P(r) of finding two monomers at distance r: Boltzmann factor Mayer f-function: difference of Boltzmann factor for two monomers at r and at λ ݂ ݎ = exp െܷ ݎ݇ Τ ܶ -1 Exclude Volume v: ݀ ݎ ݂න െ = ݒ ଷ ݎ = න 1 െ exp െܷ ݎ Τ ݇ܶ ݀ ଷ ݎ ܶ݇ Τ ݎ ܷെ exp ן ݎ ܲ
32 Criteria(判据) of polymer Solubility Gibbs free energy of mixing △G=△H-T△S Solubility occurs only when the 4G is negative △S>0 Entropy of mixing for ideal solution ASmix=k(N,In X1+N2 In X2)>0 △H??? X
3.2 Criteria (ࡔᦞ) of Polymer Solubility ¾ Solubility occurs only when the 'Gmix is negative. ' ! 0 mix S ' ' ' G H TS mix mix mix ¾ Gibbs free energy of mixing Entropy of mixing for ideal solution 1 12 2 ln ln 0 i mix ' ! S kN X N X 8 ??? 'Hmix
1. Hildebrand enthalpy of mixing (混合焓) 1-1+ 2-2- 1-2+ 1-2 1: solvent; 2: polymer △E,△E (AE/v) cohesive energy density △H △FN-2 P 12 (摩尔内聚能密度) l2 total pairs of「-2 △H nx ()(月)-6= l/2 1/2 l/2 △E 产=Vn △E1 △E2 V2 VI n[1-82] -(AE/v): solubility parameter (溶度参数)
1. Hildebrand enthalpy of mixing (ਸ❃) 9 2 1/ 2 1 2 2 / 1 2 1 mix E v H E v § · § ·§ · ' ¨ ¸ ¨ ¸¨ ¸ © ¹© ¹ © ' ¹ ' ('E/v) cohesive energy density (᪙ቄ㚊㜭ᇶᓖ) G=('E/v) 1/2 : solubility parameter (ⓦᓖ৲ᮠ) 1-1 + 2-2 1-2 + 1-2 1 1 2 2 1 2 1 2 2 mix E E E E v v v v H § ' · ' ¨ ¸ © ' ¹ ' ' 1: solvent; 2: polymer P12 2 2 2 1/2 1 1 2 1 / 1 2 m E v E v V I I ' § · § ·§ · ¨ ¸ ¨ ¸¨ ¸ © ¹© ¹ © ¹ ' > 2 @ 2 Vm 1 2 I 1 I G G 2 2 2 m n V V I1 I m m V V § · ¨ ¸ © ¹ P12 total pairs of [1-2] 2 2 n V 1 1 1 m nV V I 2 1/2 1 1 / 2 2 2 1 E v E v § · § ·§ · ¨ ¸ ¨ ¸¨ ¸ ' © ¹© ¹ © ¹ ' 1 1 nV I2 m m V V § · ¨ ¸ © ¹